Let $f:M\to M$ be a partially hyperbolic diffeomorphism. That is, there exists a continuous splitting $TM=E^u\oplus E^c\oplus E^s$ into unstable, center and stable bundles. It is well known that there exist foliations $\mathcal{W}^u$ and $\mathcal{W}^s$ tangent to $E^u$ and $E^s$, respectively.

Let's assume $f$ is dynamically coherent. That is, there exist $\mathcal{W}^{cu}$ and $\mathcal{W}^{cs}$ tangent to $E^u\oplus E^c$ and $E^s\oplus E^c$, respectively, which are subfoliated by $\mathcal{W}^u$, $\mathcal{W}^s$ and $\mathcal{W}^{c}=\mathcal{W}^{cu}\cap \mathcal{W}^{cs}$.

Let's consider a center-stable manifold $W^{cs}(x)$.

Question: When does the following hold: $W^{cs}(x)=\bigcup_{y\in W^c(x)}W^s(y)$? Is it always true, or do we need some extra assumption for it?

Another choice is to ask for $W^{cs}(x)=\bigcup_{y\in W^s(x)}W^c(y)$. Just the previous one looks more natural.

1 Answer
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No, it is in general not true. An example is the one recently constructed by Rodriguez Hertz-Rodriguez Hertz-Ures. In that example (non-dynamically coherent in dimension 3) the union of stable manifolds through a center manifold is strictly contained in the center-stable manifold (its boundary is a strong stable manifold). You can see a drawing in this slides of M.A.Rodriguez Hertz. Also, you can look at these notes I wrote (page 21) were this possibility is also discussed.

The question of wether such example can exist for a transitive partially hyperbolic diffeomorphism is open (see the paper by Bonatti-Wilkinson for more discussion).

If your question was more devoted to understanding if the union of stable manifolds is contained in a center-stable one, see the paper of Burago-Ivanov in JMD (Proposition 3.1). For results in higher dimensions, see the papers by Gogolev, Carrasco and Bonhet.